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While both specifications may be able to explain the movie, I think there might be a fourth game that explains the game the best.
While not the moral of the story that Batman wants the Joker or Gotham City to learn he tells the audience and the Joker that people are not all evil — there is some good in Gotham , adding an additional parameter to the game can easily induce a cooperation solution.
If the people aboard both ships believe that the cost of dying is probabilistic, then they would cooperate as long as the benefit to cooperating trumped both the probabilistic chance of punishment and the likelihood that the other ship decides to detonate.
Thus, each ship has to calculate the value for morality m minus the cost of punishment P c and the cost of the other ship defecting P dx d where x is the other actor.
The cost of opponent defection and punishment are the same the value of survival , so the equation reduces to:.
Each equation must still be greater than the value of Survival Minus the Cost of morality such that for Ship A or B with appropriate substitution :.
As M and S vary in relation to each then, the threshold for the joint probabilities to make a preference in strategy changes. Note, if the value for M is zero for the observed actor and survival is infinitesimal greater than 0, detonation dominates.
Likewise, if we build in opponent morality into the probability of opponent detonation, and opponent morality is zero, then the cost of morality becomes too high.
Obviously, the game becomes increasingly complicated as parameters are made endogenous. The actors must determine a subjective probability of the game being false: that is, the probability of punishment is not 1.
I would venture forth the assumption that the probability of punishment is inversely proportional to the probability that Batman exists with some modification based on his likelihood of failing to save people — but that is just me.
There are a few other situations in this movie that are prone to game theoretic applications and I have not exhausted the ways in which this game could be modeled.
However, I think this calls for a new villian in the third movie of the trilogy: The Game Theorist. Much like the riddler, but deadlier and requiring Batman to use mathematics to fight crime.
As Mike noted above, we had a lengthy conversation about this scene in the movie. Some points that I thought were particularly interesting: 1.
It played on the notion of trust. It was hard for them to trust each other not to take the detonator and simply blow up the other ship even if the majority did not agree with the decision.
So not only did they centralize authority by giving the guards sole control over the detonator and the choice to use it, at the very end the one prisoner threw the detonator out the window, making it impossible for anyone to blow up the other ship.
While in one sense it appears killing oneself is irrational, centralizing authority to ensure the outcome you want is realized is strategic and rational in that situation.
In the civilian ship, there was a completely different situation. It appears as though everyone trusted one another, so if they had a majority rule vote, they had no reason to believe anyone would disobey the result.
Although in the end no one could take personal responsibility for blowing up the other ship I discuss more about that below , they all trusted each other enough to believe they would abide by the rule.
The lesson: a group of people must trust one another in order for them to choose democratic decision rules; otherwise, they will centralize and let others make the decision for them.
It was particularly interesting that the civilians on the ferry felt comfortable to vote to blow up the prisoner ferry, but no one would take personal responsibility for it.
This reminds me of Congress, where members diffuse responsibility amongst themselves so that none of them can take personal responsibility for a piece of legislation.
I really did. The chairman bent my arm. First of all, I would like to propose something: There should be special movie screenings for political scientists.
I say this because there were 10 of us from our department in our group, and we had a tough time not geeking out immediately afterward about the game theoretical and otherwise perhaps-more-interesting-to-political-scientists aspects of this film.
I saw a few grad students from the Math department at the showing we went to, as well, so maybe they could join us in the special academia-centered screenings of future movies.
This is an excellent post, and as you mention, there are plenty of things for political scientists and game theorists of all stripes to think about in The Dark Knight.
There are two major professions in the game for players to choose from: Sorcerer and Slayer. Once the character reaches a certain level, it can be transferred to become a Destroyer.